Digital Lesson Geometric Sequences and Series An infinite

An infinite sequence is a function whose domain is the set of positive integers.

A sequence is geometric if the ratios of consecutive terms are the same. 2,

The nth term of a geometric sequence has the form an = a 1

Example: Find the 9 th term of the geometric sequence 7, 21, 63, .

The sum of the first n terms of a sequence is represented by summation

The sum of a finite geometric sequence is given by 5 + 10 +

The sum of the terms of an infinite geometric sequence is called a geometric

Example: Find the sum of The sum of the series is Copyright © by

Graphing Utility: Find the first 5 terms of the geometric sequence an = 2(1.

Slides: 10

Download presentation

Digital Lesson Geometric Sequences and Series

An infinite sequence is a function whose domain is the set of positive integers. a 1, a 2, a 3, a 4, . . . , an, . . . terms The first three terms of the sequence an = 2 n 2 are a 1 = 2(1)2 = 2 a 2 = 2(2)2 = 8 finite sequence a 3 = 2(3)2 = 18. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

A sequence is geometric if the ratios of consecutive terms are the same. 2, 8, 32, 128, 512, . . . geometric sequence The common ratio, r, is 4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3

The nth term of a geometric sequence has the form an = a 1 rn - 1 where r is the common ratio of consecutive terms of the sequence. a 1 = 15 15, 75, 375, 1875, . . . a 2 = 15(5) a 3 = 15(52) a 4 = 15(53) The nth term is 15(5 n-1). Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4

Example: Find the 9 th term of the geometric sequence 7, 21, 63, . . . a 1 = 7 an = a 1 rn – 1 = 7(3)n – 1 a 9 = 7(3)9 – 1 = 7(3)8 = 7(6561) = 45, 927 The 9 th term is 45, 927. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5

The sum of the first n terms of a sequence is represented by summation notation. upper limit of summation index of summation lower limit of summation Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6

The sum of the terms of an infinite geometric sequence is called a geometric series. If |r| < 1, then the infinite geometric series a 1 + a 1 r 2 + a 1 r 3 +. . . + a 1 rn-1 +. . . has the sum Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8

Graphing Utility: Find the first 5 terms of the geometric sequence an = 2(1. 3)n. end value variable beginning value List Menu: Graphing Utility: Find the sum upper limit List Menu: variable lower limit Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10